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Title Numerical simulations of magnetic resonance elastography using finite element analysis with a linear heterogeneousviscoelastic model

Author(s) Tomita, Sunao; Suzuki, Hayato; Kajiwara, Itsuro; Nakamura, Gen; Jiang, Yu; Suga, Mikio; Obata, Takayuki; Tadano,Shigeru

Citation Journal of visualization, 21(1), 133-145https://doi.org/10.1007/s12650-017-0436-4

Issue Date 2018-02

Doc URL http://hdl.handle.net/2115/68305

Rights(URL) https://creativecommons.org/licenses/by/4.0/

Type article

File Information 10.1007_s12650-017-0436-4.pdf

Hokkaido University Collection of Scholarly and Academic Papers : HUSCAP

REGULAR PAPER

Sunao Tomita • Hayato Suzuki • Itsuro Kajiwara • Gen Nakamura •

Yu Jiang • Mikio Suga • Takayuki Obata • Shigeru Tadano

Numerical simulations of magnetic resonanceelastography using finite element analysis with a linearheterogeneous viscoelastic model

Received: 7 October 2016 /Accepted: 19 May 2017 / Published online: 10 June 2017� The Author(s) 2017. This article is an open access publication

Abstract Magnetic resonance elastography (MRE) is a technique to identify the viscoelastic moduli ofbiological tissues by solving the inverse problem from the displacement field of viscoelastic wave propa-gation in a tissue measured by MRI. Because finite element analysis (FEA) of MRE evaluates not only theviscoelastic model for a tissue but also the efficiency of the inversion algorithm, we developed FEA forMRE using commercial software called ANSYS, the Zener model for displacement field of a wave insidetissue, and an inversion algorithm called the modified integral method. The profile of the simulated dis-placement field by FEA agrees well with the experimental data measured by MRE for gel phantoms.Similarly, the value of storage modulus (i.e., stiffness) recovered using the modified integral method withthe simulation data is consistent with the value given in FEA. Furthermore, applying the suggested FEA to ahuman liver demonstrates the effectiveness of the present simulation scheme.

Keywords Magnetic resonance elastography � Elastogram � Viscoelasticity � Finite element analysis � Liver

1 Introduction

Magnetic resonance elastography (MRE) (Muthupillai et al. 1995, 1996), which consists of magneticresonance imaging (MRI) and a time harmonic vibration system whose frequency is synchronized with theMRI pulse sequence, is a non-invasive medical diagnostic technique commonly used to diagnose lesions.This system measures the displacement field of a viscoelastic wave in response to an excitation on the

S. Tomita � H. Suzuki � I. Kajiwara (&) � S. TadanoDivision of Human Mechanical Systems and Design, Graduate School of Engineering, Hokkaido University,Kita 13, Nishi 8, Kita-ku, Sapporo, Hokkaido 060-8628, JapanE-mail: ikajiwara@eng.hokudai.ac.jpTel.: ?81 11 706 6390

G. NakamuraDepartment of Mathematics, Faculty of Science, Hokkaido University, Kita 10, Nishi 8, Kita-ku, Sapporo,Hokkaido 060-0810, Japan

Y. JiangDepartment of Applied Mathematics, Shanghai University of Finance and Economics, 777 GuoDing Road,Shanghai 200433, People’s Republic of China

M. SugaCenter for Frontier Medical Engineering, Chiba University, 1-33 Yayoicho, Inage -ku, Chiba-shi, Chiba 263-8522, Japan

T. ObataNational Institute of Radiological Sciences, 4-9-1 Anagawa, Inage-ku, Chiba-shi, Chiba 263-8555, Japan

J Vis (2018) 21:133–145https://doi.org/10.1007/s12650-017-0436-4

surface of the human body. Then solving the inverse problem for a linear viscoelastic partial differentialequation (PDE) with respect to the displacement field recovers the storage model of human organs andtissues, which can approximately model the MRE-measured displacement field. Hereafter, we refer to thisviscoelastic PDE as the PDE model. This recovery procedure is usually called an inversion, and the stiffnessmap obtained in the MRE experiment is called an elastogram. Case studies of the inversion algorithm haveinvolved the liver (Huwart et al. 2006; Klatt et al. 2007; Asbach et al. 2010), brain (Klatt et al. 2007; Sacket al. 2008; Green et al. 2008; Zhang et al. 2011), and breast tissues (Sinkus et al. 2005, 2007).

Previous studies have demonstrated the potential of finite element analysis (FEA) to evaluate theinversion algorithm as well as validated the mathematical model of MRE, the recovered storage modulus,and the excitation conditions. Van Houten et al. (1999, 2001) employed a subzone technique algorithm toreconstruct the elastic field by minimizing the difference between the set of measured displacement fieldsand those computed with FEA (Van Houten et al. 1999, 2001), while Chen et al. (2005, 2006, 2008) used anelastic PDE as a PDE model. Other studies described the damping of waves using subzone MRE techniquesemploying FEA based on the Rayleigh model (McGarry and Van Houten 2008; Van Houten et al. 2011;Petrov et al. 2014) or the poroelastic model (Perrinez et al. 2009, 2010).

Additionally, FEA has been applied to the viscoelastic equation (Atay et al. 2008; Clayton et al. 2011).Atay et al. demonstrated the reliability of their MRE reconstruction scheme by comparing the reconstructedvalue to the true value used in FEA, while Clayton et al. computed the viscoelastic waves inside a mousebrain to evaluate the accuracy of one-dimensional (1D) and three-dimensional (3D) inversion algorithms.However, both these studies assumed that the viscoelastic moduli are homogeneous. Ammari proposedoptimization-based approach for viscoelastic modulus reconstruction method and the reconstruction methodwas verified using two-dimensional (2D) heterogeneous FEA (Ammari et al. 2015a, b). Furthermore, toinvestigate the potential of MRE for the atherosclerotic plaque, Thomas-Seale et al. developed viscoelasticFEA models to compute shear modulus fields of idealized atherosclerotic plaques (Thomas-Seale et al.2011, 2016). Also, the inversion algorithms were validated based on FEA of cuboids with cylindrical inserts(Hollis et al. 2016a, b). To date, MRE algorithms have yet to be fully evaluated for 3D heterogeneousviscoelastic inversion.

This study aims to evaluate 3D numerical simulations using commercial software called ANSYS(ANSYS Incorporated, 2011) based on the Zener-type PDE model and to test its efficiency using aninversion called the modified integral method (Nakamura et al. 2008; Jiang and Nakamura 2011). Weselected commercial software to publicly share the results of FEA for MRE. Additionally, we conductedMRE measurements in agarose gel phantoms with a micro MRI (0.3 T) (Tadano et al. 2012) in Sect. 3.1. Acomparison of the simulated results by FEA to the results of the MRE measurements shows that the storagemodulus in FEA is recovered with a high accuracy in Sect. 3.2. Furthermore, we applied this inversion to theMRE measurement data of a human liver, and then the measured wave field obtained by FEA of the humanliver model was compared with the recovered viscoelastic moduli in Sect. 3.3. Finally, we assessed theaccuracy of the FEA model of the liver and the storage modulus recovered from the MRE measurementdata. It should be noted that the simulation by FEA, inversion by the modified integral method, and the MREmeasurement are linked to each other when evaluating the efficiency of MRE inversion algorithm, and thatFEA plays a key role in that link. Furthermore, MRE can be applied to other rheological studies on theviscoelastic properties of soft materials.

2 Materials and methods

2.1 Zener-type PDE model

To obtain the vibration properties of a biological tissue, a linear viscoelastic PDE was used to describe thedisplacement field of a wave in a tissue (Christensen and Richard 1982). The FEA model based on the Zenermodel was validated by comparing MRE data to FEA simulated data (Sect. 3.1). Although several vis-coelastic PDEs exist, this study used the Zener-type model (three-element Maxwell type model) because itssimulated wave images are similar to those obtained from MRE measured data (Jiang and Nakamura 2011).Figure 1 schematically depicts the 1D version of this model, where l0 and l1 are the spring constants, g1 isthe dashpot viscosity, and s1 = g1/l1 is the relaxation time. The stress–strain relation of this model isexpressed by:

134 S. Tomita et al.

r ¼Z t

0

2Gðt � sÞ deds

dsþ I

Z t

0

Kðt � sÞ dDds

ds; ð1Þ

where r is the Cauchy stress, t is the past time, e is the deviatoric strain, D is the volumetric strain, and I isthe identity matrix; G(t) and K(t) are the shear and bulk-relaxation moduli, which are described by:

GðtÞ ¼ l0 þ l1 expð�t=s1Þ; ð2Þ

KðtÞ ¼ 2

3

1þ m1� 2m

l0 þ l1 expð�t=s1Þ½ �; ð3Þ

where m is Poisson’s ratio. For the time harmonic vibration, Eq. (1) becomes:

r ¼ 2ðG0 þ iG00Þe expðiðxt þ dÞÞ þ ðK 0 þ iK 00ÞD expðiðxt þ dÞÞ; ð4Þ

where G0 and G00 are the storage and loss moduli, respectively. K0 and K00 are the storage and loss bulkmoduli, respectively, and x and d are the frequency and phase angle, respectively. The storage modulus andloss modulus are defined by:

G0 ¼ l0 þl1ðxg1Þ2

l21 þ ðxg1Þ2ð5Þ

G00 ¼ l21ðxg1Þl21 þ ðxg1Þ2

ð6Þ

2.2 Modified Stokes equation and modified integral method

Poisson’s ratio m is close to � because the tissue is nearly incompressible, which means that K0 � G0.Asymptotic analysis with respect to the large scaling parameter K0/G0 indicates that the displacement fieldu can be approximated as a solution for the following boundary value problem (7) or the modified Stokesequation (Jiang and Nakamura 2011; Jiang et al. 2011; Ammari et al. 2007), which is expressed as:

r � ½2ðG0 þ iG00ÞeðuÞ� � rpþ qx2u ¼ 0;r � u ¼ 0;u ¼ f ;omu :¼ ½2ðG0 þ iG00ÞeðuÞ � p�m ¼ 0;

8>><>>:

ð7Þ

where q is the tissue density, which can be taken as that of water (Fung 1993). p denotes the pressure fromthe longitudinal wave, and e is the linear strain tensor defined by:

eijðuÞ ¼1

2

oui

oxjþ ouj

oxi

� �: ð8Þ

Furthermore, the first two parts of Eq. (7) are considered in a domain that could be a human body or aphantom X. In contrast, the last two parts of Eq. (7) give the mixed type boundary condition with thetraction zero boundary condition on part of the boundary of X with outer unit normal n and the displacementboundary condition with a given displacement f on the rest of the boundary where the vibration is given. Itshould be noted that the displacement field u is a complex vector.

1 spring:

0

1 dash pot:

Fig. 1 Zener-type model

Numerical simulations of magnetic resonance elastography 135

If the storage and loss moduli are homogeneous, then applying the curl operator to Eq. (7) removespressure p in Eq. (7). Thus, w = r 9 u can be given as:

ðG0 þ iG00ÞDwþ qx2w ¼ 0; ð9Þ

whereD denotes the Laplacian. Because applying the curl operator to umay amplify the noise, we need to reducethe effect of noise included in u (Farahani and Kowsary 2010; Murio 1987). Herein the mollification method(Jiang and Nakamura 2011; Jiang et al. 2011; Ammari et al. 2007) was used for reducing the effect of noise.

Taking the complex conjugate of Eq. (9) gives:

ðG0 � iG00ÞDwþ qx2 �w ¼ 0 ð10Þ

Additionally, integrating the inner product with w over a test domain R yields:

G0 � iG00 ¼ �qx2

RRwj j2dxR

Rw�Dwdx

!; ð11Þ

where ‘�’ denotes the inner product. Due to the unique continuation property of the solution, the denominatorcannot vanish (Jiang and Nakamura 2011). Because the usual algebraic method does not integrate over R,Dw may vanish. This is an advantage of using the formula (11).

Equation (11), which can compute G0 - iG00 in R, is called the modified integral method (Jiang andNakamura 2011). In particular,

G0 ¼ �qx2Re

RRwj j2dxR

Rw � Dwdx

!; ð12Þ

where Re() denotes the real part in the bracket.Applying the Zener model (Jiang and Nakamura 2011) the following three important aspects were

checked carefully. First, R must be at least half of the wavelength. Second, if the heterogeneity of the tissueis sufficiently smooth and the wavelength is sufficiently small, the tissue in a small test domain R can beassumed to be homogenous. Finally, even when there is discontinuity in G0 - iG00, the modified integralequation method still produces satisfactory results.

2.3 MRE experiment with micro MRI

To validate the numerical simulation, the simulated wave image was compared with the MRE measuredwave image obtained by 0.3 T micro MRI (the Compact MRI series, MR Technology, Inc., Tsukuba, Japan).Figure 2 shows the micro MRI. The sample for the MRE measurement was a block agarose gel phantom(100 9 70 9 55 mm) placed in the micro MRI. A longitudinal wave was generated using an electrodynamic generator (C-5010 D-master, Asahi Factory Corp., Tokyo, Japan) as shown in Fig. 2. The wavepropagated to the sample through a bar comprised of glass fiber reinforced plastics (GFRP). The diameter ofthe bar head was 8 mm. The motion of the nuclear spin induced by the local movement of a tissue phantomin a gradient magnetic field induces a phase shift h at position x, which is given by:

hðxÞ ¼ cZ t0þN=F

t0

uðt; xÞ � GðtÞdt; ð13Þ

where G is the magnetic field gradient, F is the excitation frequency, u is displacement fields, c is thegyromagnetic ratio of characteristic of the nuclear isochromat, and N is the number of cycles. Equation (13)can compute the displacement fields u from the phase shift h.

2.4 FEA modeling

Figure 3 shows the FEA model and its boundary conditions. A 3D FEA model of the tissue phantom(100 9 70 9 55 mm) was created and analyzed to obtain its complex displacement fields using ‘harmonicanalysis’ in ANSYS (Version 14.0). The steady-state response of the model to sinusoidal excitation wascalculated by ‘harmonic analysis’. The model had eight node elements uniformly distributed, and eachelement measured 1.25 9 1.25 9 1.25 mm. To obtain appropriate wave fields, ten elements per a wave-length are typically necessary. The created FEA model satisfied this condition. All degrees of freedom of the

136 S. Tomita et al.

nodes on the bottom surface (an x–y slice) were fixed (Fig. 3). The nodes included in the center circular8-mm-diameter region on the y–z plane were vertically excited in the x direction at frequency f (=62.5, 125,and 250) [Hz] and 0.5 mm amplitude. The degrees of freedom of the other nodes were not restricted. Thestorage and loss moduli were computed as their true values of the inverse algorithm using Eqs. (5) and (6)when the mass density q, Poisson’s ratio m, spring constants l0 = l1, and relaxation time s1 were 1000 kg/m3, 0.499 (nearly incompressible), 7.5 kPa, and 0.025 s, respectively. The complex displacement fields u inresponse to the excitation were analyzed with this model and the above conditions.

The original Zener model with the strain–stress relation given by Eq. (1) was used to simulate thedisplacement field data instead of the modified Stokes model because the Stokes model is an approximationof this Zener model when Poisson’s ratio is very close to 0.5. In this study, this assumption was only usedfor the inversion. Mollification of the modified integral method was employed for reducing the effect ofnoise when necessary.

Fig. 2 Experimental arrangement for MRE with micro MRI

Excitation

Fixed sectionx

z

y

FEA model ofphantom

o

MRI scansection

100mm 70mm

55mm

Excitation area

0.75mm

Fig. 3 FEA model [100 mm (x) 9 70 mm (y) 9 55 mm (z)] of the tissue phantom and boundary conditions

Numerical simulations of magnetic resonance elastography 137

2.5 FEA of the heterogeneous model

Next we evaluated the accuracy of the inversion algorithm with respect to the heterogeneous viscoelasticmodel (Fig. 4). A columnar phantom with a diameter of 10, 15, or 20 mm was embedded in the blockmodel. All columnar phantoms had heights of 55 mm. Eight node elements were used and the elementnumbers of each columnar phantom were 1056, 2992, and 5456. The viscoelastic parameters of thebackground gels were the same as those of the homogeneous model. The columnar materials had a springconstant of l0 = l1 = 15 kPa, while the other parameters were the same as the background parameters. Thestorage moduli G0 of the background and columnar gels were computed from Eq. (5) as 15 and 30 kPa,respectively.

2.6 Validation of the inversion scheme

The inversion method was validated using both homogeneous and heterogeneous models. The viscoelasticmoduli were recovered by applying the inversion scheme based on the modified integral method to thedisplacement field data computed by FEA of the homogeneous model at 62.5, 125, and 250 Hz. Dataprocessing to recover the viscoelastic modulus fields from the FEA results was performed with programswritten with MATLAB R2013a (Mathworks). For the inversion, the number of points for the numericalintegration in each direction, Nx, Ny, and Nz, was set to 3, and the computations were conducted with the dataobtained by FEA (coordinate resolution = 1.25 mm). The numerical integration must have an appropriatenumber of points for reliable simulations of the MRE inversion scheme. If too few points are used in thenumerical integration, a highly accurate storage modulus cannot be recovered. Thus, the parameters must beset appropriately.

To compare the 3D inversion to the commonly used 2D inversion, the 2D inversion method was appliedto the computed displacement data on the MRI scan section (z = 32.5 mm) in Fig. 3. For the 2D inversion,the number of points for the integral computation was set to Nx = Ny = 3 and Nz = 1. Additionally, the 3Dinversion scheme was applied to the heterogeneous FEA results.

2.7 FEA modeling of a human liver

The 3D FEA of a human liver was conducted to simulate the in vivo MRE experiment of a healthy volunteer(male, age 22 years). The subject provided written informed consent, and the study was approved by theinstitutional ethics committee in the National Institute of Radiological Science. A spin-echo echo-planarimaging (SE-EPI) pulse sequence with a motion-encoding gradient (MEG) was used to visualize the shearwave pattern in the subject. The experiment was performed with a 3.0 T MRI scanner (Signa HDx; GEHealthcare) using the following parameters: field of view (FOV) = 288 9 288 mm, imagingmatrix = 64 9 64, the number of slices = 7, slice thickness = 4.5 mm, TR = 448 ms, and TE = 41.7 ms.To evaluate this experiment by FEA, the 3D shape data of a real human liver extracted from an MRI scanner

ColumnarFE modelG’ = 30kPa

Background FE modelG’ = 15kPa

z

y xo

Fig. 4 Example of a heterogeneous FEA model of a tissue phantom, including a 20-mm columnar phantom

138 S. Tomita et al.

was used to create the FEA model. The FEA model was discretized via a four-node element with ICEMCFD 14.0. ANSYS 14.0 (ANSYS Incorporated 2011) was employed assuming that the liver model washomogeneous. The interface of the passive drive (diameter 40 mm) section was excited in the normaldirection at 62.5 Hz (Fig. 5). The storage modulus was set as the mean of the measured elastogram in theregion of interest (ROI) (Fig. 10).

3 Results

3.1 Comparison to the experimental image

To evaluate the wave propagation model inside a soft tissue, Fig. 6a-1–a-3 shows the MRE displacementfield images of 1.2 wt% agarose gel at frequencies of 62.5, 125, and 250 Hz. Figure 6b-1–b-3 shows the 2Ddisplacement fields at z = 32.5 mm extracted from the 3D displacement fields computed by FEA. Thedisplacement fields were computed using a homogeneous model and the setup shown in Fig. 3. The spatialwavelength of the displacement field is the dominant parameter for the storage modulus, but the dis-placement scale is negligible. The experimental and simulated wavenumbers in the images at each fre-quency (one at 62.5 Hz, two at 125 Hz, and five at 250 Hz) agree, indicating that the spatial wavelengths arequalitatively similar. This finding validates the model employed in this study.

3.2 Validation of the MRE scheme

Figure 7 shows an example of the homogeneous numerical simulation results, where the 3D modifiedintegral method used to numerically simulate the data was applied to the displacement fields in the y-direction excited at 250 Hz and the recovered storage modulus fields G0. Table 1 shows the averages of thestorage moduli recovered from the wave fields excited at each frequency by the inversion scheme. Themaximum error between the recovered storage modulus and the measured value is 22.7% in the 2Dinversion and 2.7% in the 3D inversion. Figure 8 depicts the relations between the excitation frequency andthe recovered storage modulus. The 3D modified integral method reproduces well the theoretical storagemoduli at various frequencies.

Figure 9a, b shows the storage modulus fields and the computed wave fields in the heterogeneous FEA.Additionally, Fig. 9c depicts the recovered storage modulus fields using the computed displacement fieldsand the 3D modified integral method. It is observed from Fig. 9 that the modified integral method canreconstruct the heterogeneous elastic modulus fields identifying the difference of the modulus between thebase material and the inclusion. These results show that the recovered storage modulus fields and measuredvalues agree well, validating the effectiveness of the inversion method.

3.3 FEA of the human liver model

Figure 10 shows the magnitude with the ROI and the elastogram of the liver in a healthy volunteer. Themean storage modulus in the ROI is 1.18 kPa. Figure 11 shows the computed wave fields inside a humanliver propagating from left to right. To extract the spatial wavelength over the entire region of the liver, the2D Fourier transforms of the measured wave fields (Fig. 11b) and the FEA wave fields (Fig. 11a) werecomputed. The 2D Fourier transform yields a predominant peak wavelength of 19.5 mm, while that for the

Excitationat 62.5Hz

Fig. 5 FEA model of a human liver

Numerical simulations of magnetic resonance elastography 139

calculated data is 19.2 mm. Because the spatial wavelength of the wave fields corresponds to the storagemodulus of the object, the recovered storage modulus is considered to be accurate. This means that thedeveloped FEA can assure the reliability of the reconstruction of viscoelastic modulus by MRE algorithmbecause FEA was conducted with the reconstructed storage modulus from the measured data.

mm

(a-1)Experimental wave image

(62.5Hz excitation)

(a-2)Experimental wave image

(125Hz excitation)

(a-3)Experimental wave image

(250Hz excitation)

mm

(b-1)Computed wave image

(62.5Hz excitation)

(b-2)Computed wave image

(125Hz excitation)

(b-3)Computed wave image

(250Hz excitation)

mm

mm

0 20 40 60 800

20

40

60

mm

mm

0 20 40 60 800

20

40

60

mm

mm

0 20 40 60 800

20

40

60

mm

mm

0 20 40 60 80 1000

20

40

60

mm

mm

0 20 40 60 80 1000

20

40

60

mm

mm

0 20 40 60 80 1000

20

40

60

Fig. 6 Experimental and computed wave images excited at 62.5, 125, or 250 Hz on an MRI scan slice (z = 32.5 mm)

Fig. 7 Real parts of the computed displacement fields along the y axis and the recovered storage modulus fields (Slice 1:y = 57.5 mm, Slice 2: x = 81.25 mm, Slice 3: z = 11.25 mm) using the same parameters as Fig. 3

Table 1 Storage modulus recovered from the FEA results

Frequency (Hz) True (kPa) 3D reconstruction (kPa) 2D reconstruction (kPa)

62.5 14.5 14.3 ± 2.31 14.9 ± 10.3125 14.9 14.9 ± 0.90 17.5 ± 3.60250 15.0 15.4 ± 0.75 18.4 ± 3.31

140 S. Tomita et al.

4 Discussion

This study aims to validate the reliability of the MRE measurement by constructing a numerical simulationscheme of the forward problem and the inversion scheme. Initially, to verify the accuracy of the mathe-matical model used in MRE, FEA of the viscoelastic model was used to simulate the MRE of agarose gel at62.5-, 125-, and 250-Hz excitations with a micro-MRI system (Fig. 2). A comparison of the MRE waveimages of agarose gel phantoms obtained by micro MRI at 62.5-, 125-, and 250-Hz excitations (Fig. 6)shows that the values agree well at all excitation frequencies, confirming that the viscoelastic model set inthe forward problem simulation is appropriate to obtain the vibration properties of soft components such asan agarose gel phantom. The slight mismatch is due to difficulties modeling the vibration excitation.Regardless, these results show that the appropriate mathematical model of FEA can be used to simulatewave propagation inside soft tissues.

Next, we verified the accuracy of the inversion method by recovering the storage modulus from the FEAresults using the 3D modified integral method (Nx = Ny = Nz = 3). The storage modulus for the FEAsimulations is consistent with the recovered storage modulus as there is a less than 3% difference betweenthe two values at all excitation frequencies (Table 1; Fig. 8), demonstrating that the mathematical model ofthe inverse problem is applicable and inverse analyses are accurate.

The 3D modified integral method was used to recover the storage modulus, and the improvement overthe inversion with the 2D displacement fields was evaluated using the 2D displacement fields extracted fromthe 3D computed displacement fields. A 2D inversion (Nx = Ny = 3, Nz = 1) produces less accurate resultsbecause the wavelengths of the 2D displacement images extracted from the 3D displacement images arelarger than the true wavelength, which is consistent with previously reported results in multi-slice MREexperiments (Feng et al. 2013). Hence, the 3D inversion measures the storage modulus with a high accuracy.

To simulate MRE for detecting lesions, FEA of the heterogeneous viscoelastic model was conducted.The storage modulus fields recovered with the 3D modified integral method agree well with the measuredstorage modulus for all sizes of columnar materials (Fig. 9a, b). FEA of MRE is useful to evaluate theelastograms of heterogeneous storage modulus fields, and the 3D modified integral method can recover theheterogeneous storage modulus fields with a high accuracy. Consequently, the 3D modified integral methodcan determine the storage modulus fields with a high accuracy without changing the wave propagationdirection at the boundary section.

In addition, the 3D modified integral method was applied to evaluate the MRE measurement data of ahuman liver, and FEA of the human liver model was conducted using the geometry of a human liver and themean of the recovered storage modulus in the ROI. A comparison of the spatial wavelengths of themeasured wave fields confirms the effectiveness of FEA for computing wave propagation in a human liveras well as the accuracy of recovered storage modulus (Fig. 11b) and the calculated wave fields (Fig. 11a).Hence, FEA simulations of the MRE can help establish a mathematical model of MRE and assess theaccuracy of the inversion. In the diagnosis of liver diseases such as hepatic fibrosis, the recovered storagemodulus can be evaluated by comparing with the ideal wave fields computed with FEA.

FEA simulations of MRE can be used to construct a database of various diseases, assuring the reliabilityof the viscoelastic moduli of human organs estimated using MRE. In addition, applying FEA simulations to

0

5

10

15

20

25

0 100 200 300

Theoretical

3D modified integral method

2D modified integral method

Excitation frequency [Hz]

Stor

age

mod

ulus

[kPa

]

Fig. 8 Recovered storage moduli

Numerical simulations of magnetic resonance elastography 141

the human body geometry can evaluate wave propagation in the human body as well as the excitationconditions (e.g., actuator location, generating power of the actuator, and excitation frequencies for eachorgan), enabling efficient optimization for each tissue. Thus, the FEA simulations should improve thereliability of medical diagnoses based on MRE.

5 Conclusion

FEA simulations of MRE are introduced to evaluate the MRE measurements. FEA of the viscoelastic modelcan simulate wave propagation inside agarose gel phantom and a human liver. Both homogeneous andheterogeneous FEA simulations validate the accuracy of the inversion method, which was applied to the

mm02mm51mm01

kPa (a) True storage modulus fields

mm (b) Wave fields computed with FEA

kPa (c) Storage modulus fields recovered

with 3D modified integral method

Fig. 9 Recovered heterogeneous storage modulus fields at z = 32.5 mm

142 S. Tomita et al.

MRE measurement data of a human liver to evaluate the recovered storage modulus. The 3D modifiedintegral method is highly accurate, demonstrating that FEA simulations are useful to quantitatively evaluatea clinical diagnosis based on MRE.

Acknowledgements This study was supported by a Grant-in-Aid for the development of Advanced Measurement and AnalysisSystems (2009–2011), the Japan Science and Technology Agency, and by the Joint Research Project under the Japan–USCooperative Science Program (2012–2013), the Japan Society for the Promotion of Science. We also acknowledge theanonymous referees’ comments, which greatly improved this manuscript.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium,provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license,and indicate if changes were made.

margotsalE(b)edutinga M(a)

ROI

ROI

kPa

Fig. 10 a Magnitude and b elastogram of the liver in a healthy volunteer

(a) Computed wave fields

(b) Measured wave fields

Imaginary partReal part

Imaginary partReal part

Fig. 11 Computed and experimental wave fields propagating from left to right

Numerical simulations of magnetic resonance elastography 143

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